High Rate Single-Symbol ML Decodable Precoded DSTBCs for Cooperative Networks

1 High Rate Single-Symbol ML Decodable Precoded DSTBCs for Cooperati v e Netw orks Harshan J and B. Sunda r Rajan, Senior Member , IEEE Abstract Distributed Orthogonal Space-T ime Block Cod es (DOSTBCs) ac hiev ing full diversity order and single-sy mbol ML decod ability hav e been introdu ced recently by Y i a nd Kim for cooperati ve networks and an upperbo und on the maximal rate of such co des along with code construc tions has been presented. In this paper , we introduce a ne w class of Distributed ST BCs called S emi-orthogona l P recoded Distributed Single-Symbol Decodable S TBCs (S-P DSSDC) wherein, the source performs co-ordinate interleaving of information symbols appropriately before transmitting i t to all t he relays. It is shown that DOSTBCs are a special case of S-PDS SDCs. A special class of S- PDSSDCs having diagonal cov ariance matrix at the destination is studied and an upperbound on the maximal rate of such co des is deri ved. The bounds obtained are approximately twice l arger than that of the DOSTBCs. A systematic construction of S-PDSSDCs is presented when the number of relays K ≥ 4 . The constructed codes are sho wn to achie ve the upperbou nd on the rate when K is of the form 0 or 3 mod ulo 4. For the r est of the v alues of K , the constructed codes are shown to hav e rates higher than that of DOSTBCs. It i s shown that S-P DSSDCs cannot be constructed with any form of l inear processing at the relays when the source doesn’t perform co-ordinate interleaving of the information symbols. Simulation result sho ws that S-P DSSDCs ha ve better proba bility of error performan ce than that of DOSTBCs. Index T erms Cooperati ve div ersity , single-symbol ML decoding, distributed space-time coding, precoding. I . I N T RO D U C T I O N A N D P R E L I M I N A R I E S C oopera ti ve commun ication has been a promising m eans of ach ieving spatial diversity without the n eed of multiple anten nas at the individual no des in a wireless n etwork. T he idea is based o n the rela y channel model, where a set of distributed antennas belongin g to multiple users in th e n etwork co- operate to en code th e sign al transmitted f rom the sourc e and f orward it to th e de stination so that the req uired diversity o rder is achieved, [1]-[4]. Spatial diversity obtain ed fro m such a co- operation is refe rred to as co- operative diversity . In [5], the idea of space- time codin g d evised for po int to point co-located multiple a ntenna systems is ap plied for a wir eless relay network and is r eferred to as distributed space-time coding. T he technique inv olves a two phase pr otocol wher e, in the first This work wa s supported through grants to B.S. Rajan ; partly by the DRDO-IISc program on Advance d Researc h in Mathematic al E nginee ring, and partly by the Counci l of Scientific & Industrial Research (CSIR, India) Research Grant (22(0365) /04/EMR-II). Part of the content of this paper has been submitted to IEEE Internationa l Conferenc e on Communicati ons (ICC 2008 ). The author s are with the Department of Elect rical Communicat ion Engineerin g, Indian Institu te of Science, Bangalore-56001 2, India. Email: { harsha n,bsrajan } @e ce.iisc.ernet.in. August 10, 2018 DRAFT 2 phase, the source b roadcasts the information to the relays and in the second phase, the relays linearly process the signals received from the source and f orward them to the destinatio n such th at the sig nal at the d estination appears as a Space-T ime Block Code (STBC). Since the work of [1]-[5 ], lot o f ef forts h av e been made to generalise the various aspects of s pace-time coding propo sed for multiple anten na systems to th e co-operative setu p. One such importan t aspect is the design o f low- complexity Maximum Likelihood (ML) deco dable Distributed Spa ce-T ime Block Codes (DST BCs) - in pa rticular, the design o f Sing le-Symbol M L Decoda ble (SSD) DSTBCs. For a back groun d on SSD STBCs for MI MO systems, we refer the r eader to [6] - [1 2]. Th rough out th e p aper, we consider DSTBCs th at are ML de codable. T w o gr oup decodab le DSTBCs wer e intro duced in [13] th rough do ubling constru ction using a commuting set of matrices from field extensions. In [14], Orthogo nal Designs (ODs) and Quasi-orthog onal Designs [9] originally propo sed fo r multiple an tenna systems have been app lied to the co-oper ativ e framework. Sinc e the co-variance m atrix of additive noise at the d estination is a fun ction of (i) the rea lisation of the chan nels from the relays to the destination and (ii) the relay matrices, co mplex orth ogonal designs (except fo r 2 re lays - Alamo uti cod e) loose their SSD pro perty in the co -operative setup. In [15], DSTBCs based on co-or dinate interleaved orthogon al designs [11] have b een introdu ced which have re duced decodin g comp lexity . I n this set-up, the sou rce perfor ms co-or dinate inter leaving of informa tion symbo ls before tr ansmitting to the relays. In [16 ], low decoding c omplexity DSTBC s were prop osed using Clifford-algeb ras, wherein the relay no des are assumed to have the knowledge of th e ph ase com ponent of the sou rce-to-r elay channels. A class of fo ur-group deco dable DSTBCs was also proposed in [ 17]. Recently , in [18], Distributed Orthogon al Space-Time Codes (DOSTBCs) achieving single-sym bol d ecodability have been intr oduced f or co-o perative networks. The authors co nsidered a special class o f DOSTBCs wh ich make the cov ariance matrix of the additive no ise vector at the destination , a diagon al one an d such a class of codes was referr ed to as row mo nomial DOSTBCs. Upper bound s o n the maxim um symbol-rate (in comp lex sym bols per channel use in the secon d ph ase) o f row monom ial DOSTBCs have been derived and a systematic con struction of such codes has been proposed. The constru cted codes were sho wn to m eet the u pperbo und fo r even numb er of relays. In [20], the same a uthors have derived an upp erboun d on the symbol-r ate o f DOSTBCs when the additive noise at the destination is c orrelated and have shown that the improvement in the rate is not significant wh en compare d to the case wh en the n oise at the destination is u ncorrela ted [18]. In [1 9] and [2 0], SSD DSTBCs hav e bee n studied when th e relay nodes are assumed to know the corresp onding channel phase informa tion. An up perbou nd on the sy mbol rate for such a set up is shown to b e 1 2 which is indepen dent of the numb er of relays. In [18], [ 19] and [20] the sou rce n ode tr ansmits the informatio n symbols to all th e relay s with out any pr ocessing. On the similar lines o f [15] an d using the fr amew ork proposed in [18], in this pape r , we propose SSD DSTBCs aided by linear pr ecoding of the information vector at the source. In our set-up, we a ssume that the relay nodes do not have the knowledge of the channel fro m the sour ce to itself. In particu lar , it is shown that, co-o rdinate interleaving of informa tion sym bols at the sour ce along with the appropr iate cho ice o f rela y matrices, SSD DSTBCs with m aximal rates higher than that of DOSTBCs can b e co nstructed. The contributions of this p aper can be summarized as August 10, 2018 DRAFT 3 follows: • A new class of DSTBC s called Pre coded DSTBCs ( PDSTBCs) (Definition 1) is in troduced where th e source perfor ms co-ordin ate inte rleaving of info rmation symbols approp riately before tran smitting it to all the relays. W ithin this class, we identify codes th at are SSD and refer to them as Preco ded Distrib uted Sing le Symbol Decodable STBC s (PDSSDCs) (Definition 2 ). The well known DOSTBCs studied in [1 8] a re sho wn to be a special case o f PDSSDCs. • A set o f necessary an d sufficient c onditions on the r elay m atrices for the existence of PDSSDCs is proved (Theor em 1). • W ithin the set of PDSSDCs, a class of Semi-orthog onal PDSSDCs ( S-PDSSDC) (Definition 4) is d efined. The known DOSTBCs are shown to belo ng to the class of S-PDSSDCs. On the similar lines of [ 18], a specia l class of S- PDSSDCs ha ving a diagonal covariance matrix at the d estination is studied and are referred to as row mo nomial S-PDSSDCs. An u pperbo und on the m aximal sy mbol-rate of row mo nomial S-PDSSDCs is derived. I t is shown that, the sy mbol r ate o f r ow m onomial S-PDSSDC is upperb ounded by 2 l and 2 l +1 , w hen the number o f relay s, K is of the f orm 2 l and 2 l + 1 respectively , wh ere l is any natural num ber . The bo unds obtained are approximately twice larger than that of DOSTBCs. • A systematic co nstruction of row-monomial S-PDSSDCs is presente d when K ≥ 4 . Codes achieving the upperb ound on th e symb ol rate are constructed when K is 0 or 3 m odulo 4. F or the rest of the values of K , the constru cted S-PDSSDCs are s hown to h av e rates higher th an that of the DOSTBCs. • Precoding of information symb ols at the source has resulted in the c onstruction o f high ra te S-PDSSDCs. In this setup, the relays d o no t perfo rm co -ordin ate interleaving of th e received sy mbols. It is shown that, when th e sou rce tran smits inf ormation symbo ls to all the relays with o ut any p recodin g, an d if the relay s are allowed to perf orm linear processing of their received vector, S-PDSSDCs other than DOSTBCs cannot be co nstructed thereb y , n ecessitating the source to p erform coo rdinate interleaving of inf ormation symbols in order to construct high rate S- PDSSDCs. The remaining part of the paper is organized as follows: In Sectio n I I, along w ith th e signal model, PDSTBCs are introdu ced a nd a special class o f it called PDSSDCs is d efined. A set o f n ecessary and sufficient condition s o n the relay matrices for the existence of PDSSDCs is also derived. In Section III, S-PDSSDCs are defined and a special class of it called row-monomial S-PDSSDCs are studied . An up perbou nd on the m aximal rate o f row-monom ial S-PDSSDCs is derived. In Section IV, construction of row-monomial S-PDSSDCs is pr esented along with some examples. In Section V, we sho w that the sou rce has to necessarily perform precoding of information symbo ls in order to c onstruct hig h ra te S-PDSSDCs. The pr oblem of d esigning two-dimensio nal signal sets for the fu ll diversity of RS-PDSSDCs is discussed in Section VI along with some simulation results. Conc luding remar ks and po ssible directions for further work constitute Section VII. August 10, 2018 DRAFT 4 Notations: Th rough ou t the p aper, bo ldface letter s an d capital boldface letters are used to represent vectors and matrices respectively . For a complex ma trix X , the matrices X ∗ , X T , X H , | X | , Re X and Im X den ote, respectiv ely , the con jugate, transpose, conjug ate transpo se, determinan t, real part and ima ginary p art of X . The element in th e r th 1 row and the r th 2 column o f the matr ix X is denoted by [ X ] r 1 ,r 2 . The diagon al matr ix diag { [ X ] 1 , 1 , [ X ] 2 , 2 · · · [ X ] T ,T } constructed from the diagonal elements of a T × T ma trix X is d enoted by diag [ X ] . For complex ma trices X and Y , X ⊗ Y deno tes the ten sor p roduct of X and Y . The tenso r prod uct o f th e matrix X with itself r times where r is any positive integer is represented by X ⊗ r . The T × T id entity m atrix an d the T × T zero matrix respecti vely denoted by I T and 0 T . The magn itude of a complex n umber x , is d enoted b y | x | and E [ x ] is used to denote the expectation of the rando m variable x. A circular ly symmetric complex Gaussian ra ndom vector, x with mean µ and covariance matrix Γ is denoted by x ∼ C S C G ( µ, Γ ) . T he set of all integers, the real number s and th e complex number s are r espectiv ely , deno ted by Z , R an d C and j is used to represent √ − 1 . The set of all T × T complex diagona l matrice s is d enoted by D T and a sub set of D T with strictly positiv e diagon al elements is d enoted by D + T . I I . P R E C O D E D D I S T R I B U T E D S PAC E - T I M E C O D I N G A. Signal mode l The wireless network co nsidered as shown in Figure 1 consists of K + 2 nod es each having single a ntenna which are place d randomly and indepen dently acco rding to some distribution. There is one sour ce node and one destination node. All th e other K nodes are relay s. W e den ote th e channel from the sour ce n ode to the k th relay as h k and the ch annel fro m th e k th relay to the de stination n ode as g k for k = 1 , 2 , · · · , K . The fo llowing assumptions are made in our mo del: • All the nodes a re sub jected to half duplex con straint. • Fading coefficients h k , g k are i.i. d C S C G (0 , 1) with coherence time inter val of atleast N and T respectively . • All the nodes a re syn chronized at the symbol le vel. • Relay no des do not ha ve th e kn owledge of fade coef ficients h k . • Destination kn ows the fade coef ficients g k , h k . The sour ce is equipp ed with a N length complex vector from the code book S = { s 1 , s 2 , s 3 , · · · , s L } consisting of inform ation vectors s l ∈ C 1 × N such that E  s l s H l  = 1 for all l = 1 , · · · , L . The sourc e is also eq uipped with a pair of N × N matrices P and Q called precoding matrices. Every transmission from the sour ce to the destination comprises of two p hases. When th e source n eeds to tran smit a n info rmation vector s ∈ S to the de stination, it generates a ne w vector ˜ s as, ˜ s = sP + s ∗ Q (1) such that E h ˜ s ˜ s H i = 1 and bro adcasts the vector ˜ s to all the K relay s (but not to the destination ). The received vector at the k th relay is given by r k = √ P 1 N h k ˜ s + n k , f or all k = 1 , 2 , · · · , K where n k ∼ C S C G (0 , I N ) is the additive noise at the k th relay and P 1 is th e total power used at the source node e very ch annel use. In the secon d phase, all the relay nodes are scheduled to transmit T leng th vecto rs to the destina tion simultaneously . Each r elay August 10, 2018 DRAFT 5 is equ ipped with a fixed p air of N × T re ctangular ma trices A k , B k and is allowed to lin early proc ess the rece i ved vector . The k th relay is s chedu led to tran smit t k = s P 2 T (1 + P 1 ) N { r k A k + r ∗ k B k } . (2) where P 2 is the total power used at each relay every chan nel use in the second phase. T he vector r eceiv ed at the destination is gi ven b y y = K X k =1 g k t k + w (3) where w ∼ C S C G (0 , I T ) is th e add iti ve n oise at the destination. Using (2 ) in (3), y can be written as y = s P 1 P 2 T (1 + P 1 ) N gX + n where • n = q P 2 T (1+ P 1 ) N h P K k =1 g k { n k A k + n ∗ k B k } i + w . • The equ i valent chan nel g is gi ven by [ g 1 g 2 · · · g K ] ∈ C 1 × K . • Every codeword X ∈ C K × T is of the form, X = h [ h 1 ˜ sA 1 + h ∗ 1 ˜ s ∗ B 1 ] T [ h 2 ˜ sA 2 + h ∗ 2 ˜ s ∗ B 2 ] T · · · [ h K ˜ sA K + h ∗ K ˜ s ∗ B K ] T i T . Definition 1: Th e collection C of K × T cod ew ord matrices shown below , where s runs over a codebook S , C =  h [ h 1 ˜ sA 1 + h ∗ 1 ˜ s ∗ B 1 ] T [ h 2 ˜ sA 2 + h ∗ 2 ˜ s ∗ B 2 ] T · · · [ h K ˜ sA K + h ∗ K ˜ s ∗ B K ] T i T ff (4) is called the Precoded Distributed Space-T ime Block code (PDSTBC) which i s determined by the set { P , Q , A k , B k } . Remark 1: From (4), every codeword of a PDSTB C includes random variables h k for all k = 1 , 2 , · · · K . Even though , h k can take any complex value, since the destinatio n knows the ch annel set { h 1 , h 2 , · · · h K } fo r every codeword use, the cardinality of C is equal to the cardin ality of S . The p roperties o f the PDSTBC will depend on the set { P , Q , A k , B k } alone b ut not on the realisation of the ch annels h k ’ s. In this pa per , on the similar lines of [18], we derive conditions on the set { P , Q , A k , B k } such that the PDSTBC in (4) is SSD for any values of { h 1 , h 2 , · · · h K } . In othe r words, the derived cond itions are su ch that irre spectiv e o f the realisation o f h k ’ s, the PDSTBC in (4) is SSD. The covariance matr ix R ∈ C T × T of the noise vector n is gi ven b y R = P 2 T (1 + P 1 ) N " K X k =1 | g k | 2 n A H k A k + B H k B k o # + I T . (5) The Maxim um Likelihood (ML) decoder decodes to a vector ˆ s where ˆ s = a r g min s ∈S " y − s P 1 P 2 T (1 + P 1 ) N gX # R − 1 " y − s P 1 P 2 T (1 + P 1 ) N gX # H August 10, 2018 DRAFT 6 = arg min s ∈S " − 2 Re s P 1 P 2 T (1 + P 1 ) N gXR − 1 y H ! + P 1 P 2 T (1 + P 1 ) N gXR − 1 X H g H # . W ith the above deco ding m etric, we give a definition fo r a SSD distributed space- time b lock code which also includes DOSTBCs studied in [18]. Definition 2: A PDSTBC, X in v ariables x 1 , x 2 , · · · x N is called a Preco ded Dis tributed Single-Symbol Deco dable STBC (PDSSDC), if it satisfies the following co nditions, • The entries o f the k th row o f X are 0, ± h k ˜ x n , ± h ∗ k ˜ x ∗ n or multiples of these by j where j = √ − 1 for any complex variable h k . The co mplex variables ˜ x n for 1 ≤ n ≤ N a re the com ponents of the tran smitted vector ˜ s where ˜ s = [ ˜ x 1 ˜ x 2 · · · ˜ x N ] . • The matrix X satisfies the e quality XR − 1 X H = N X i =1 W i with [ W i ] k,k = | h k | 2  υ (1) i,k | x iI | 2 + υ (2) i,k | x iQ | 2  (6) where each W i is a K × K matrix with its non z ero entries being f unctions of x iI , x iQ and h k for all k = 1 , 2 , · · · , K an d υ (1) i,k , υ (2) i,k ∈ R . W e study th e pro perties of th e relay matrices A k , B k and the p recoding ma trices P an d Q such that th e vectors transmitted simu ltaneously from all th e relay s appear a s a PDSSDC at the destination. Certain pro perties of the relay matrices have been studied in the context o f DOSTBCs in [1 8]. W e recall some of the definitions and properties used in [1 8] so as to stu dy the pr operties of the r elay matr ices of a PDSSDC. A ma trix is said to b e colum n (row) monom ial, if there is atmo st one no n-zero entry in e very column (row) of it. Lemma 1: The relay matrices A k and B k of a PDSSDC satisfy the follo wing conditio ns, • The entrie s of A k and B k are 0, ± 1 , ± j . • A k and B k cannot have n on-zero s at the same po sition. • A k , B k and A k + B k are co lumn mono mial matrices. Pr oo f: The pr oof is on th e similar lin es of the proof for L emma 1 in [ 18]. Lemma 2: If A , C , D ∈ C N × N and s = [ x 1 , x 2 , · · · , x N ] ∈ C 1 × N , with each x i = x iI + j x iQ , then sAs H + s Cs T + s ∗ Ds H = N X i =1 f i ( x iI , x iQ ) (7) where f i ( x iI , x iQ ) is a complex valued fu nction o f the variables x iI and x iQ if and only if A , C + C T , D + D T ∈ D N . Pr oo f: Refer to the proof of Le mma 2 in [21]. Using the r esults o f Lemma 2, in th e follo wing Theorem, we provide a set of necessary and suf ficient co nditions on the matrix set { P , Q , A k , B k } such that a PDSTBC X with the above matrix set is a PDSSDC. Theor em 1: A PDSTBC X is a PDSSDC if a nd o nly if the relay matrices A k , B k satisfy the fo llowing conditions, August 10, 2018 DRAFT 7 (i) For 1 ≤ k 6 = k ′ ≤ K , Υ 1 A k R − 1 A H k ′ Υ H 2 + Π ∗ 1 A ∗ k ′ R − 1 A T k Π T 2 ∈ D N for 8 > > < > > : Υ 1 = Υ 2 = P and Π 1 = Π 2 = Q ; Π 2 = Υ 1 = P and Π 1 = Υ 2 = Q ; Π 1 = Υ 2 = P and Π 2 = Υ 1 = Q ; (8) Υ ∗ 1 B k R − 1 B H k ′ Υ T 2 + Π 1 B ∗ k ′ R − 1 B T k Π H 2 ∈ D N for 8 > > < > > : Υ 1 = Υ 2 = Q and Π 1 = Π 2 = P ; Π 2 = Υ 1 = Q and Π 1 = Υ 2 = P ; Π 1 = Υ 2 = Q and Π 2 = Υ 1 = P . (9) (ii) For 1 ≤ k , k ′ ≤ K , Π ∗ h B k R − 1 A H k ′ + A ∗ k ′ R − 1 B T k i Υ H ∈ D N , for 8 > > < > > : Υ = P and Π = Q ; Υ = P and Π = P ; Υ = Q and Π = Q ; (10) Π h A k R − 1 B H k ′ + B ∗ k ′ R − 1 A T k i Υ T ∈ D N , for 8 > > < > > : Υ = Q and Π = P ; Υ = P and Π = P ; Υ = Q and Π = Q . (11) (iii) For 1 ≤ k ≤ K , A k R − 1 A H k + B ∗ k R − 1 B T k = diag [ D 1 ,k , D 2 ,k , · · · , D N,k ] . (12) where D n,k ∈ R for all n = 1 , 2 , · · · N . Pr oo f: Refer to the proof of Le mma 3 in [21]. Theorem 1 pr ovides a set of necessary and sufficient co nditions on the relay matrices A k , B k and the pre - coding matrices P and Q such th at, X is a PDSSDC. T he ma trices, Υ 1 A k R − 1 A H k ′ Υ H 2 + Π ∗ 1 A ∗ k ′ R − 1 A T k Π T 2 , Υ ∗ 1 B k R − 1 B H k ′ Υ T 2 + Π 1 B ∗ k ′ R − 1 B T k Π H 2 , Π ∗ h B k R − 1 A H k ′ + A ∗ k ′ R − 1 B T k i Υ H and Π h A k R − 1 B H k ′ + B ∗ k ′ R − 1 A T k i Υ T in the conditio ns of (8) - (11) need to be diagon al. This implies that the above matrices can also be 0 N . The DOSTBCs studied in [18] are a spec ial class of PDSSDCs since the relay m atrices o f D OSTBCs ( Lemma 1 , [ 18 ]) satisfy the conditions o f Theore m 1. In p articular, the necessary and suf ficient condition s on the relay matr ices of DOSTBCs as shown in Lem ma 1 of [ 18 ] can be obtained fro m the necessary and sufficient con ditions of PDSSDCs by mak ing P = I N , Q = 0 N and D N = 0 N in (8) - (11). A PDSSDC, X in variables x 1 , x 2 , · · · x N can be written in th e fo rm of a lin ear d ispersion code [22] as X = P N j =1 x iI Φ iI + x iQ Φ iQ where Φ iI , Φ iQ ∈ C K × T are called the weigh t matrices of X . W ithin the class of PDSSDCs, we consider a special set of codes c alled Un itary PDSSDCs defined as, Definition 3: A PDSSDC, X is called a Unitary PDSSDC, if the weigh t matrices o f X satisfies the following condition s, Φ iI Φ H iI , Φ iQ Φ H iQ ∈ D + K for all i = 1 , 2 , · · · N . Remark 2: W e caution the rea der to no te the difference between the definition for a Un itary PDSSDC for cooper ati ve networks and the definition for a Unitary SSD co de fo r MIMO systems [1 2]. For better clarity , we recall the definition for a Unitary SS D code designed for MIMO systems. A SSD STBC, ˜ X in variables x 1 , x 2 , · · · x N when August 10, 2018 DRAFT 8 written in the form of a linear disper sion cod e is given b y ˜ X = P N j =1 x iI ˜ Φ iI + x iQ ˜ Φ iQ , w here ˜ Φ iI , ˜ Φ iQ ∈ C K × T are called the weight matr ices o f ˜ X . The design ˜ X is said to b e a unitar y SSD if ˜ Φ iI ˜ Φ H iI = I K for all i = 1 , 2 , · · · N . The difference between the two d efinitions can b e ob served as th e defin ition for a unitary SSD STBC is a special case of the definition for a unitary PDSSDC. It c an be verified that DOSTBCs belon g to the class o f Unitary PDSSDCs. In the rest of the paper, we co nsider only unitary PDSSDCs. Howe ver , it is to be no ted that the class of non- unitary PDSSDCs is not empty . A class of low decod ing com plexity DSTBCs called Precoded C oord inate In terleaved Orth ogona l Design (PCIOD) h as b een introdu ced in [ 15] where in the auth ors h av e pro posed a design, X P C I O D for a n etwork with 4 relays which is SSD (Example 1 of [15]). It can be ob served tha t th e proposed code X P C I O D giv en in (13) belong s to the class of non-u nitary PDSSDCs. Since we con sider o nly unitary PDSSDCs, through out the paper a PDSSDC is meant unitary PDSSDC. X P C I O D = 2 6 6 6 6 6 4 h 1 ˜ x 1 h 1 ˜ x 2 0 0 − h ∗ 2 ˜ x ∗ 2 h ∗ 2 ˜ x ∗ 1 0 0 0 0 h 3 ˜ x 3 h 3 ˜ x 4 0 0 − h ∗ 4 ˜ x ∗ 4 h ∗ 4 ˜ x ∗ 3 3 7 7 7 7 7 5 . (13) The pre coding matrices, P and Q r equired at the source to con struct X P C I O D are P = 1 2 2 6 6 6 6 6 4 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 3 7 7 7 7 7 5 ; Q = 1 2 2 6 6 6 6 6 4 1 0 − 1 0 0 1 0 − 1 − 1 0 1 0 0 − 1 0 1 3 7 7 7 7 7 5 . V arious class of single-sym bol decoda ble STBCs for coop erative networks are captured in Figure 2 which is first partitioned in to two sets depending on whe ther the codes are u nitary o r non- unitary (Definition 3) . T he class of PDSSDCs are shown to be a subset of the class of SSD codes f or cooper ativ e networks. The set of unitary distributed SSD codes are sho wn to contain the DOSTBCs and the S- PDSSDCs (Definition 4). An example of a code which belongs to the class of no n-unitar y D istributed SSD cod es b ut not to the class of PDSSDCs is given below , X DS S D C =   Re ( h 1 x 1 ) + j Im ( h 1 x 2 ) − Re ( h 1 x 2 ) + j Im ( h 1 x 1 ) Re ( h 2 x 2 ) + j Im ( h 2 x 1 ) Re ( h 2 x 1 ) − j Im ( h 2 x 2 )   . (14) A 1 = 1 2 2 4 1 1 1 − 1 3 5 ; B 1 = 1 2 2 4 1 − 1 − 1 − 1 3 5 ; A 2 = 1 2 2 4 1 1 1 − 1 3 5 and B 2 = 1 2 2 4 − 1 1 1 1 3 5 . From the above matrices, it can be verified that, R − 1 is a scaled identity matrix . Therefo re, XR − 1 X H = R − 1 XX H and XX H is given by , XX H =   | h 1 | 2 P 2 i =1 | x i | 2 ( h ∗ 1 h 2 − h ∗ 2 h 1 ) P 2 i =1 | x i | 2 ( h ∗ 2 h 1 − h ∗ 1 h 2 ) P 2 i =1 | x i | 2 | h 2 | 2 P 2 i =1 | x i | 2   . August 10, 2018 DRAFT 9 I I I . S E M I - O RT H O G O N A L P D S S D C From the definition of a PDSSDC (Defin ition 2),  XR − 1 X H  k,k ′ for any k 6 = k ′ can be non-zer o. i. e, the k th and the k ′ th row of a PDSSDC X , n eed not satisfy th e equa lity  XR − 1 X H  k,k ′ = 0 , but  XR − 1 X H  k,k ′ must be a com plex linear comb ination o f se veral terms with each term be ing a function o f in -phase and quad rature compon ent of a single in formation variable. Thro ugh out th e pap er , the k th and the k ′ th row of a PDSSDC are referred to as R -orth ogonal if  XR − 1 X H  k,k ′ = 0 . Similarly , the k th and the k ′ th row are r eferred to as R -non- orthog onal if  XR − 1 X H  k,k ′ 6 = 0 . In th is paper, we identify a special c lass of PDSSDCs wh ere e very r ow of X is R -non- orthogo nal to atmost o ne of its ro ws and we formally defin e it as, Definition 4: A PDSSDC is said to be a Semi-o rthogo nal PDSSDC (S-PDSSDC) if every row of a PDSSDC is R -non- orthogo nal to atmost o ne of its ro ws. From the above de finition, it can be ob served that DOSTBCs are a proper subclass of S-PDSSDCs since every r ow of DOSTBC is R -orthog onal to every other r ow . The definitio n o f a S-PDSSDC implies that the set of K rows can be partition ed in to atleast ⌈ K 2 ⌉ gr oups such that every g roup has atmost two rows. The co-variance matrix, R in ( 5) is a functio n of (i) the r ealisation of the channe ls from the relays to the destination and (ii) the r elay matrices, A k , B k . In general, R may not be diagonal in which c ase the construction of S-PDSSDCs is not straight for ward. On the similar lines of [18], we consider a subset of S-PDSSDCs wh ose covariance matrix is diago nal a nd refer to such a subset as ro w mo nomial S-PDSSDCs ( RS-PDSSDCs). It can be proved that the relay matrices of a RS-PDSSDC are row mo nomial if an d o nly if the cor respondin g covariance matrix is diagon al (ref er to Theorem 1 of [18]) . The r ow mon omial p roperty of th e relay matrices implies that every row o f a RS-PDSSDC contains the variables ± h k ˜ x n and ± h ∗ k ˜ x ∗ n atmost o nce for all n such that 1 ≤ n ≤ N . A. upperbo und on the symbol-rate of RS -PDSSDCs In this subsection , we d eriv e an u pperbo und on the rate of RS-PDSSDCs in symbols p er cha nnel use in the second phase i.e an u pperbo und on N T . T owards that end, proper ties of the relay matric es A k , A k ′ , B k and B k ′ of RS-PDSSDC are studied wh en the rows co rrespond ing to the indices k and k ′ are (i) R -ortho gonal and (ii) R -non- orthogo nal. For the former c ase, the p roperties of A k , A k ′ , B k and B k ′ have been stud ied in [18]. If k a nd k ′ represent the in dices of the rows of a RS-PDSSDC th at are R - orthog onal, then the cor respondin g rela y matrices A k , A k ′ , B k and B k ′ satisfies the following con ditions (i) A k and A k ′ are co lumn d isjoint and (ii) B k and B k ′ are column disjoint. (Lemma 3 o f [18]) i.e ., the matrices A k and A k ′ cannot con tain non- zero entr ies on th e same columns simultaneo usly . The ab ove result imp lies, A k A H k ′ = 0 N and B ∗ k B T k ′ = 0 N . (15) In ord er to add ress the latter case, consider the 2 × 2 matrix Ξ as gi ven below , Ξ =   h k ˜ x i h k ′ ˜ x m h k  h k ′ ♣   August 10, 2018 DRAFT 10 where h k , h k ′ are complex ran dom variables. The complex variables ˜ x i and ˜ x m are the co mponen ts of the tra nsmitted vector ˜ s (as in (1)) wher e ˜ s = [ ˜ x 1 ˜ x 2 · · · ˜ x N ] . In p articular, th e complex variables ˜ x i and ˜ x m are of th e fo rm, ˜ x i = ± x γ  ± j x λ  and ˜ x m = ± x δ  ± j x µ  where • γ , λ, δ and µ are positive in tegers such that 1 ≤ γ , λ, δ , µ ≤ N and atmost any two of these inte gers can be equal. • The subscrip t  deno tes either I (in-ph ase co mponen t) or Q (quadratu re com ponent) o f a v ariable and •  , ♣ are indetermin ate co mplex variables which can tak e v alues o f the form ± ˜ x n or ± ˜ x ∗ n such that 1 ≤ n ≤ N . For example, if N = 4, ˜ x i and ˜ x m can p ossibly be x 2 I + j x 3 Q and x 3 I + j x 4 Q respectively . In Lemm a 3, we investigate v arious choices on the indeter minate variables  and ♣ such tha t  Ξ H Ξ  1 , 2 is a complex linear combin ation of sev eral terms with eac h term be ing a fu nction of in-ph ase and quadr ature componen ts of a single in formation variable. In gen eral, the re al variables x γ  , x λ  , x δ  and x µ  can app ear in ˜ x i and ˜ x m with ar bitrary signs. Wit h out lo ss of g enerality , we assume th at ˜ x i and ˜ x m are g i ven by ˜ x i = x γ  + j x λ  and ˜ x m = x δ  + j x µ  . (16) Howe ver , the results of Lemma 3 will continu e to hold ev en if the variables x γ  , x λ  , x δ  and x µ  appear in ˜ x i and ˜ x m with any arbitra ry signs. Since a RS-PDSSDC takes variables on ly o f the f orm ± h k ˜ x n , ± h ∗ k ˜ x ∗ n and ev ery row of a RS-PDSSDC c ontains the variables ± h k ˜ x n and ± h ∗ k ˜ x ∗ n atmost once, we have the fo llowing re strictions on the choice of the indeterminate variables  and ♣ that (i) the ind eterminate  cannot take the variable ˜ x i and variables of the form ˜ x ∗ n for all n = 1 , 2 , · · · N a nd (ii) the indeterminate ♣ cannot take the variable ˜ x m and variables o f the fo rm ˜ x ∗ n for all n = 1 , 2 , · · · N . Lemma 3: If there exists a solution on the ch oice of  and ♣ su ch that h Ξ H Ξ i 1 , 2 = f 1 ( x δI , x δQ , h k , h k ′ ) + f 2 ( x γ I , x γ Q , h k , h k ′ ) + f 3 ( x λI , x λQ , h k , h k ′ ) + f 4 ( x µI , x µQ , h k , h k ′ ) 6 = 0 , (17) then on ly one o f the fo llowing is true, (i) δ = γ and µ = λ . (ii) δ = λ a nd µ = γ . where f i ( x β I , x β Q , h k , h k ′ ) is a complex valued fu nction o f the variables, x β I , x β Q , h k , h k ′ for all i = 1 , 2 , · · · 4 and β = µ, λ, γ , δ. Pr oo f: Refer to the proof of Le mma 4 in [21]. Similarly , it can be shown that, the re sults of L emma 3 h olds true even if th e matrix Ξ is of the form,   h ∗ k ˜ x ∗ i h ∗ k ′ ˜ x ∗ m h ∗ k  h ∗ k ′ ♣   . W e use th e resu lts of Lem ma 3 to study the pr operties of the rela y matrices of a RS-PDSSDC. Lemma 4: Let A k and A k ′ be the relay matrice s of a RS-PDSSDC, X . If h A k A H k ′ i i,m is a no n zero entry for August 10, 2018 DRAFT 11 i 6 = m , then the p recoding matrices at the source P and Q ar e such that ˜ x iI , ˜ x iQ , ˜ x mI and ˜ x mQ ∈ { x nI , x nQ , x n ′ I , x n ′ Q } with (18) ˜ x iI , ˜ x iQ ∈ { x n  , x n ′  } a nd ˜ x mI , ˜ x mQ ∈ { x n  , x n ′  } for some n 6 = n ′ where 1 ≤ n, n ′ ≤ N an d the subscript  represen ts either I or Q . Pr oo f: Refer to the proof of Le mma 5 in [21]. Lemma 5: Let B k and B k ′ be th e relay matrices of a RS-PDSSDC, X . If  B ∗ k B T k ′  i,m is a n on zero entry f or i 6 = m , precoding matrices at the source P a nd Q are su ch th at ˜ x iI , ˜ x iQ , ˜ x mI and ˜ x mQ ∈ { x nI , x nQ , x n ′ I , x n ′ Q } with (19) ˜ x iI , ˜ x iQ ∈ { x n  , x n ′  } a nd ˜ x mI , ˜ x mQ ∈ { x n  , x n ′  } for some n 6 = n ′ where 1 ≤ n, n ′ ≤ N an d the subscript  represen ts either I or Q . Pr oo f: The resu lt can b e proved on the similar lines of th e pr oof f or Lemm a 4. Cor olla ry 1: For a RS-PDSS DC, if h A k A H k ′ i i,m is non -zero, then so is h A k A H k ′ i m,i . Pr oo f: Follows fr om the proo f for Lemm a 3 and Lemma 4. From the definition of a PDSSDC (Definition 2), non-zero entries o f the k th row contains variables of the form ± h k ˜ x n , ± h ∗ k ˜ x ∗ n or multip les of these by j . The refore,  XX H  k,k = | h k | 2 h ˜ sA k A H k ˜ s H + ˜ s ∗ B k B H k ˜ s T i + h k h k h ˜ sA k B H k ˜ s T i + h ∗ k h ∗ k h ˜ s ∗ B k A H k ˜ s H i = N X i =1 | h k | 2  ω (1) i,k | x iI | 2 + ω (2) i,k | x iQ | 2  where ω (1) i,k , ω (2) i,k ∈ R + for all k = 1 , 2 , · · · , K . From the results of L emma 1 in [ 10], we ha ve A k A H k + B ∗ k B T k = diag [ E 1 ,k , E 2 ,k , · · · E n,k ] (20) where E n,k are strictly positi ve r eal numb ers. Lemma 6: Let k an d k ′ represent the indices o f the rows of a RS-PDSSDC, that are R -no n-ortho gonal, then the correspo nding relay matrices A k , B k , A k ′ and B k ′ satisfy the follo wing conditio ns, • h A k A H k ′ i i,i =  B ∗ k B T k ′  i,i = 0 f or all i = 1 , 2 · · · N . • A k A H k ′ and B ∗ k B T k ′ are both column and row mono mial m atrices. • A k A H k ′ + B ∗ k B T k ′ is co lumn and r ow mo nomial matrix. • The nu mber of no n-zero entries in A k A H k ′ + B ∗ k B T k ′ is even. • The m atrices ˜ A k,k ′ and ˜ B k,k ′ giv en by ˜ A k,k ′ = h A T k A T k ′ i T and ˜ B k,k ′ =  B T k B T k ′  T satisfy the following August 10, 2018 DRAFT 12 inequality : Rank h ˜ A k,k ′ ˜ A H k,k ′ + ˜ B ∗ k,k ′ ˜ B T k,k ′ i ≥    2 m if N = 2 m and 2 m + 2 if N = 2 m + 1 (21) where m is a positive in teger . Pr oo f: Refer to the proof of Le mma 7 in [21]. Using the properties of rela y m atrices A k , A k ′ , B k and B k ′ of a RS-PDSSDC cor respondin g to two different rows that are ( i) R -orthog onal and (ii) R -no n-orth ogonal, an upper bound on the maximum r ate, N T is der i ved in the following th eorem. Theor em 2: The symbol-rate of a RS -PDSSDC satis fies the in equality : Rate = N T ≤ 8 > > > > > < > > > > > : 2 l if N = 2 m, K = 2 l 2 l +1 if N = 2 m, K = 2 l + 1 2 m +1 ( m +1) l if N = 2 m + 1 , K = 2 l 4 m +2 (2 m +2) l +2 m +1 if N = 2 m + 1 , K = 2 l + 1 . (22) where l and m are positive integers. Pr oo f: Refer to the proof of Th eorem 1 in [21]. I V . C O N S T RU C T I O N O F R S - P D S S D C S In this section, we construct RS-PDSSDCs when the n umber of relays K ≥ 4 . The constru ction provides co des achieving the up perbou nd in (22) when (i) N and K a re multiples of 4 and ( ii) N is a multiple of 4 and K is 3 modulo 4. F or the r est of the values of N and K , codes meeting the up perbou nd are not kno wn. In p articular, for values of N < 4 a nd any K , th e autho rs a re not aware of RS-PDSSDCs with rates highe r than that o f row monomial DOSTBCs. The fo llowing co nstruction provides RS-PDSSDCs with rates h igher than that of row monom ial DOSTBCs when N ≥ 4 and K ≥ 4 . W e first p rovide the constru ction of the p recoding matr ices P an d Q an d then p resent the co nstruction of R S-PDSSDCs. A. Construction o f pr ecoding matrices P and Q Let Γ , Ω ∈ C 4 × 4 be given by Γ = 1 2 2 6 6 6 6 6 4 1 0 − j 0 0 1 0 − j 0 1 0 j 1 0 j 0 3 7 7 7 7 7 5 and Ω = 1 2 2 6 6 6 6 6 4 1 0 j 0 0 1 0 j 0 − 1 0 j − 1 0 j 0 3 7 7 7 7 7 5 . Let N = 4 y + a , wh ere a can take values of 0 , 1 , 2 an d 3 and y is any positi ve integer . For a gi ven v alue of a and y , the p recodin g matrices P and Q at the source are co nstructed as, P = 2 4 Γ ⊗ I y 0 4 y × a 0 a × 4 y I a 3 5 ; Q = 2 4 Ω ⊗ I y 0 4 y × a 0 a × 4 y 0 a 3 5 . August 10, 2018 DRAFT 13 Example 1 : For N = 6 , we h a ve y = 1 and a = 2 . Following the ab ove construction metho d, prec oding matrices P an d Q are gi ven b y , P = 1 2 2 6 6 6 6 6 6 6 6 6 6 4 1 0 − j 0 0 0 0 1 0 − j 0 0 0 1 0 j 0 0 1 0 j 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 3 7 7 7 7 7 7 7 7 7 7 5 ; Q = 1 2 2 6 6 6 6 6 6 6 6 6 6 4 1 0 j 0 0 0 0 1 0 j 0 0 0 − 1 0 j 0 0 − 1 0 j 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 7 7 7 7 7 7 7 7 7 7 5 . B. Construction o f RS-PDSS DCs Throu gh o ut th is su bsection, we denote a RS-PDSSDC f or K relays with N variables as X ( N , K ) . Constru ction of RS-PDSSDCs is divided in to three cases dependin g on the values of N and K . Case 1: N = 4 y and K = 4 x : In this case, we construct RS-PDSSDCs in the following 4 steps. Step (i) : Let U x 1 , x 2 be a 2 × 2 Alamouti design in complex variables x 1 , x 2 as given below , U x 1 , x 2 =   x 1 x 2 − x ∗ 2 x ∗ 1   . (23) Using th e design in (23), co nstruct a 4 × 4 design, Ω m in 4 complex variables ˜ x 4 m +1 , ˜ x 4 m +2 , ˜ x 4 m +3 and ˜ x 4 m +4 as shown below for all m = 0 , 1 , · · · y − 1 . Ω m = 2 4 U ˜ x 4 m +1 , ˜ x 4 m +2 U ˜ x 4 m +3 , ˜ x 4 m +4 U ˜ x 4 m +3 , ˜ x 4 m +4 U ˜ x 4 m +1 , ˜ x 4 m +2 3 5 = 2 6 6 6 6 6 4 ˜ x 4 m +1 ˜ x 4 m +2 ˜ x 4 m +3 ˜ x 4 m +4 − ˜ x ∗ 4 m +2 ˜ x ∗ 4 m +1 − ˜ x ∗ 4 m +4 ˜ x ∗ 4 m +3 ˜ x 4 m +3 ˜ x 4 m +4 ˜ x 4 m +1 ˜ x 4 m +2 − ˜ x ∗ 4 m +4 ˜ x ∗ 4 m +3 − ˜ x ∗ 4 m +2 ˜ x ∗ 4 m +1 3 7 7 7 7 7 5 where ˜ x 4 m +1 = x (4 m +1) I + j x (4 m +4) Q ; ˜ x 4 m +2 = x (4 m +2) I + j x (4 m +3) Q ; ˜ x 4 m +3 = x (4 m +1) Q + j x (4 m +4) I ; ˜ x 4 m +4 = x (4 m +2) Q + j x (4 m +3) I . Step (ii) : Le t H , ∆ an d Θ ∈ C K × K giv en by H = diag { h 1 , h 2 , · · · , h K } , ∆ = diag { 1 , 0 , 1 , 0 , · · · 0 } and Θ = diag { 0 , 1 , 0 , 1 , · · · 1 } where h 1 , h 2 , · · · h K are complex variables and ∆ , Θ are such that ∆ + Θ = I K . Using H , ∆ and Θ , construct a diago nal matrix, G as G = H ∆ + H ∗ Θ . Step (iii) : Using Ω m , co nstruct a 4 x × 4 x matr ix X m giv en by Ω m ⊗ I ⊗ ( x − 1) 2 for each m = 0 , 1 , · · · y − 1 . Step (iv) : A RS-PDSSDC, X ( N , K ) is c onstructed u sing X m and G as X ( N , K ) = G [ X 0 X 1 · · · X y − 1 ] where the m atrix [ X 0 X 2 · · · X y − 1 ] is obtained by ju xtaposing the m atrices X 0 , X 1 , · · · , X y − 1 . Example 2 : For N = 4 and K = 4 , we hav e x = y = 1 . Following Step (i) to Step (iv) in the above constru ction, August 10, 2018 DRAFT 14 we have G = diag { h 1 , h ∗ 2 , h 3 , h ∗ 4 } an d X 0 = Ω 0 . Hen ce X (4 , 4) is given by , X (4 , 4) = 2 6 6 6 6 6 4 h 1 ˜ x 1 h 1 ˜ x 2 h 1 ˜ x 3 h 1 ˜ x 4 − h ∗ 2 ˜ x ∗ 2 h ∗ 2 ˜ x ∗ 1 − h ∗ 2 ˜ x ∗ 4 h ∗ 2 ˜ x ∗ 3 h 3 ˜ x 3 h 3 ˜ x 4 h 3 ˜ x 1 h 3 ˜ x 2 − h ∗ 4 ˜ x ∗ 4 h ∗ 4 ˜ x ∗ 3 − h ∗ 4 ˜ x ∗ 2 h ∗ 4 ˜ x ∗ 1 3 7 7 7 7 7 5 . (24) where ˜ x 1 = x 1 I + j x 4 Q ; ˜ x 2 = x 2 I + j x 3 Q ; ˜ x 3 = x 1 Q + j x 4 I and ˜ x 4 = x 2 Q + j x 3 I . The variables ˜ x 1 , ˜ x 2 , · · · ˜ x 4 are obtained using the p recoding m atrices P and Q as given in (1). The p recoding matrice s P and Q are constructed as in Subsection IV -A. The r elay specific matrices A k , B k for the RS-PDSSDC in (2 4) are as gi ven b elow , A 1 = 2 6 6 6 6 6 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 3 7 7 7 7 7 5 ; B 2 = 2 6 6 6 6 6 4 0 1 0 0 − 1 0 0 0 0 0 0 1 0 0 − 1 0 3 7 7 7 7 7 5 ; A 3 = 2 6 6 6 6 6 4 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 3 7 7 7 7 7 5 and B 4 = 2 6 6 6 6 6 4 0 0 0 1 0 0 − 1 0 0 1 0 0 − 1 0 0 0 3 7 7 7 7 7 5 . B 1 = A 2 = B 3 = A 4 = 0 4 . Example 3 : For N = 4 and K = 8, we have y = 1 an d x = 2 . Following th e constructio n p rocedu re in Case 1, X 0 = Ω 0 ⊗ I 2 and X (4 , 8) = G X 0 . Th erefore, X (4 , 8) is given by . X ( 4 , 8) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 h 1 ˜ x 1 h 1 ˜ x 2 h 1 ˜ x 3 h 1 ˜ x 4 0 0 0 0 − h ∗ 2 ˜ x ∗ 2 h ∗ 2 ˜ x ∗ 1 − h ∗ 2 ˜ x ∗ 4 h ∗ 2 ˜ x ∗ 3 0 0 0 0 h 3 ˜ x 3 h 3 ˜ x 4 h 3 ˜ x 1 h 3 ˜ x 2 0 0 0 0 − h ∗ 4 ˜ x ∗ 4 h ∗ 4 ˜ x ∗ 3 − h ∗ 4 ˜ x ∗ 2 h ∗ 4 ˜ x ∗ 1 0 0 0 0 0 0 0 0 h 5 ˜ x 1 h 5 ˜ x 2 h 5 ˜ x 3 h 5 ˜ x 4 0 0 0 0 − h ∗ 6 ˜ x ∗ 2 h ∗ 6 ˜ x ∗ 1 − h ∗ 6 ˜ x ∗ 4 h ∗ 6 ˜ x ∗ 3 0 0 0 0 h 7 ˜ x 3 h 7 ˜ x 4 h 7 ˜ x 1 h 7 ˜ x 2 0 0 0 0 − h ∗ 8 ˜ x ∗ 4 h ∗ 8 ˜ x ∗ 3 − h ∗ 8 ˜ x ∗ 2 h ∗ 8 ˜ x ∗ 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . Case 2: N = 4 y and K = 4 x + a for a = 1 , 2 and 3 : In this case, a RS-PDSSDC is constructed in two step s as given below . Step(i) : Construct a RS-PDSSDC for parameter s N = 4 y and K = 4( x + 1) as giv en in Case 1. Step(ii) : Drop the last 4 − a rows of the RS-PDSSDC constru cted in Step ( i). Example 4 : When N = 4 and K = 6, the parameters a , x an d y ar e 2, 1 and 1 respectively . As given in Case 2, a RS-PDSSDC for N = 4 and K = 8 is constructed and the last 2 rows of the design are drop ped. The code August 10, 2018 DRAFT 15 X (4 , 6) is as g iv en below . X ( 4 , 6) = 2 6 6 6 6 6 6 6 6 6 6 4 h 1 ˜ x 1 h 1 ˜ x 2 h 1 ˜ x 3 h 1 ˜ x 4 0 0 0 0 − h ∗ 2 ˜ x ∗ 2 h ∗ 2 ˜ x ∗ 1 − h ∗ 2 ˜ x ∗ 4 h ∗ 2 ˜ x ∗ 3 0 0 0 0 h 3 ˜ x 3 h 3 ˜ x 4 h 3 ˜ x 1 h 3 ˜ x 2 0 0 0 0 − h ∗ 4 ˜ x ∗ 4 h ∗ 4 ˜ x ∗ 3 − h ∗ 4 ˜ x ∗ 2 h ∗ 4 ˜ x ∗ 1 0 0 0 0 0 0 0 0 h 5 ˜ x 1 h 5 ˜ x 2 h 5 ˜ x 3 h 5 ˜ x 4 0 0 0 0 − h ∗ 6 ˜ x ∗ 2 h ∗ 6 ˜ x ∗ 1 − h ∗ 6 ˜ x ∗ 4 h ∗ 6 ˜ x ∗ 3 3 7 7 7 7 7 7 7 7 7 7 5 . Case 3: N = 4 y + b an d K = 4 x + a where b = 1 , 2 , 3 and a = 0 , 1 , 2 , 3 : In this case, RS-PDSSDCs ar e constructed in the following 3 steps. Step (i) : Con struct a RS-PDSSDC, X (4 y , 4 x + a ) for par ameters N = 4 y and K = 4 x + a as in Case 2 using the first 4 y variables. Step (ii) : Construct a DOSTBC, X ′ ( b, 4 x + a ) with p arameters N = b an d K = 4 x + a using the last b variables as in [18]. Step (iii) : Th e RS-PDSSDC, X ( N , K ) is given by jux taposing X (4 y , 4 x + a ) and X ′ ( b, 4 x + a ) as shown below , X ( N , K ) =  X (4 y , 4 x + a ) X ′ ( b, 4 x + a )  . Example 5 : When N = 6 and K = 8 , the p arameters b, a, x and y are respec ti vely g iv en by 2, 0, 2 and 1. As in Step (i), con struct X (4 , 8) as exp lained in Case 1 wh ich is g i ven below , X ( 4 , 8) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 h 1 ˜ x 1 h 1 ˜ x 2 h 1 ˜ x 3 h 1 ˜ x 4 0 0 0 0 − h ∗ 2 ˜ x ∗ 2 h ∗ 2 ˜ x ∗ 1 − h ∗ 2 ˜ x ∗ 4 h ∗ 2 ˜ x ∗ 3 0 0 0 0 h 3 ˜ x 3 h 3 ˜ x 4 h 3 ˜ x 1 h 3 ˜ x 2 0 0 0 0 − h ∗ 4 ˜ x ∗ 4 h ∗ 4 ˜ x ∗ 3 − h ∗ 4 ˜ x ∗ 2 h ∗ 4 ˜ x ∗ 1 0 0 0 0 0 0 0 0 h 5 ˜ x 1 h 5 ˜ x 2 h 5 ˜ x 3 h 5 ˜ x 4 0 0 0 0 − h ∗ 6 ˜ x ∗ 2 h ∗ 6 ˜ x ∗ 1 − h ∗ 6 ˜ x ∗ 4 h ∗ 6 ˜ x ∗ 3 0 0 0 0 h 7 ˜ x 3 h 7 ˜ x 4 h 7 ˜ x 1 h 7 ˜ x 2 0 0 0 0 − h ∗ 8 ˜ x ∗ 4 h ∗ 8 ˜ x ∗ 3 − h ∗ 8 ˜ x ∗ 2 h ∗ 8 ˜ x ∗ 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (25) According to Step (ii), con struct a DOSTBC [18], X ′ (2 , 8) as sho wn below , X ′ (2 , 8) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 h 1 ˜ x 5 h 1 ˜ x 6 0 0 0 0 0 0 − h ∗ 2 ˜ x ∗ 6 h ∗ 2 ˜ x ∗ 5 0 0 0 0 0 0 0 0 h 3 ˜ x 5 h 3 ˜ x 6 0 0 0 0 0 0 − h ∗ 4 ˜ x ∗ 6 h ∗ 4 ˜ x ∗ 5 0 0 0 0 0 0 0 0 h 5 ˜ x 5 h 5 ˜ x 6 0 0 0 0 0 0 h ∗ 6 ˜ x ∗ 6 h ∗ 6 ˜ x ∗ 5 0 0 0 0 0 0 0 0 h 7 ˜ x 5 h 7 ˜ x 6 0 0 0 0 0 0 − h ∗ 8 ˜ x ∗ 6 h ∗ 8 ˜ x ∗ 5 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (26) August 10, 2018 DRAFT 16 A RS-PDSSDC X (6 , 8 ) is constructed by juxtaposing th e desig ns in (25) and (26) as sho wn below , X (6 , 8) =  X ( 4 , 8) X ′ (2 , 8)  . C. Comparison of the Symbol-rates of RS -PDSSDCs and r ow-monomia l DOS TBCs For a given value of N , K such th at N ≥ 4 an d K ≥ 4 , we pro posed a method of con structing a RS-PDSSDC, X ( N , K ) w ith a m inimum value of T . The minimum values of T p rovided in our con struction is listed below against the co rrespond ing values of N and K . Against every value of T for RS-PDSSDCs, the correspond ing value of T for row mono mial DOSTBC is provided with in th e br aces. (i) N ev en, K even : T ≥ 8 > > > > > < > > > > > : 4 xy (8 xy ) if N = 4 y , K = 4 x. 4 xy + 4 x (8 xy + 4 x ) if N = 4 y + 2 , K = 4 x. 4 xy + 4 y (8 xy + 4 y ) if N = 4 y , K = 4 x + 2 . 4 xy + 4 y + 4 x + 2 (8 x y + 4 y + 4 x + 2) if N = 4 y + 2 , K = 4 x + 2 . (ii) N ev en, K odd : T ≥ 8 > > > > > < > > > > > : 4 xy + 4 y (8 xy + 4 y ) if N = 4 y , K = 4 x + 1 . 4 xy + 4 y + 4 x + 2 (8 x y + 4 x + 4 y + 2) if N = 4 y + 2 , K = 4 x + 1 . 4 xy + 4 y (8 xy + 8 y ) if N = 4 y , K = 4 x + 3 . 4 xy + 4 y + 4 x + 4 (8 x y + 8 y + 4 x + 4) if N = 4 y + 2 , K = 4 x + 3 . (iii) N o dd, K ev en : T ≥ 8 > > > > > < > > > > > : 4 xy + 4 x (8 xy + 4 x ) if N = 4 y + 1 , K = 4 x. 4 xy + 4 y + 4 x + 2 (8 x y + 4 x + 4 y + 2) if N = 4 y + 1 , K = 4 x + 2 . 4 xy + 8 x (8 xy + 8 x ) if N = 4 y + 3 , K = 4 x. 4 xy + 4 y + 8 x + 4 (8 x y + 4 y + 8 x + 4) if N = 4 y + 3 , K = 4 x + 2 . (iv) N odd , K odd : T ≥ 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : 4 xy + 4 y + 4 x + 1 ( max (8 xy + 4 x + 2 y + 1 , 8 xy + 4 y + 2 x + 1)) i f N = 4 y + 1 , K = 4 x + 1 . 4 xy + 4 y + 4 x + 3 ( max (8 xy + 6 y + 4 x + 3 , 8 xy + 8 y + 2 x + 2)) i f N = 4 y + 1 , K = 4 x + 3 . 4 xy + 8 x + 4 y + 3 ( max (8 xy + 6 x + 4 y + 3 , 8 xy + 8 x + 2 y + 2)) if N = 4 y + 3 , K = 4 x + 1 . 4 xy + 4 y + 8 x + 8 ( max (8 xy + 8 x + 6 y + 6 , 8 xy + 8 y + 6 x + 6)) i f N = 4 y + 3 , K = 4 x + 3 . From the above comp arison, it can be observed that, for a g i ven value o f N and K , a RS-PDSSDC, X ( N , K ) is con structed with a smaller value of T compar ed to tha t of a row monom ial DOSTBC, ther e by pr oviding higher values of the sy mbol- rate, N T . I n par ticular , when N is a multiple of 4 and K is of the form 0 or 3 mod ulo 4, row monom ial DOSTBCs need dou ble the numb er of channel uses in the second phase com pared to that o f RS-PDSSDCs. It can also be observed that improvement in th e v alues of T for a RS-PDSSDC is not significant August 10, 2018 DRAFT 17 when K an d N are b oth odd. V . O N T H E C O N S T RU C T I O N O F S - P D S S D C S W I T H O U T P R E C O D I N G A T T H E S O U R C E The existence of high rate S-PDSSDCs has been shown in the preced ing sections, when the sou rce p erforms co-ord inate interleaving of infor mation symbo ls befor e bro adcasting it to all the relays. In this setup, the relays do not per form coordinate interleaving of their receiv ed sy mbols. One obviou s question that needs to be answered is, whether lin ear p rocessing of the received symbols at the relay s a lone is suffi cient to construc t S-PDSSDCs wh en the source doesn’t per form coordinate in terleaving of info rmation sym bols. In other word s, is coo rdinate interlea ving of the information symbols at the source nece ssary to con struct PDSSDCs ? . Th e answer is, yes. In the rest of this section , we show that PDSSDCs cannot be constru cted by linear processing of the received symbols at the relays when the so urce transmits the inf ormation sym bols to all the relays with ou t pr ecoding. T owards th at end, let the k th relay be equipped with a pair of m atrices, A k and B K ∈ C N × T which perf orm linear processing on the recei ved vector . Ex cluding the additi ve noise component, the recei ved vector at the k th relay is h k s = [ h k x 1 , h k x 2 , · · · h k x N ] wher e x i ’ s ar e inform ation symbols and h k is any com plex number . The m atrices A k , B K ∈ C N × T act on the vector h k s to generate a vector of the form, h k sA k + h ∗ k s ∗ B k (27) From (27), the n on zero entries of h k sA k + h ∗ k s ∗ B k contains complex variables of the fo rm, ± x, ± x ∗ or multiples of these by j where j = √ − 1 and Re ( x ) , I m ( x ) ∈ { Re ( h k x n ) , Im ( h k x n ) | 1 ≤ n ≤ N } . (28) T o b e p recise, Re ( h k x n ) and Im ( h k x n ) are given by h kI x nI − h kQ x nQ and h kI x nQ + h kQ x nI respectively . Th e above vector can also contain line ar combin ation o f the s pecified ab ove comp lex variables. From Definition 2, non-zer o entries of the k th row of a PDSSDCs are of the form ± h k ˜ x n , ± h ∗ k ˜ x ∗ n where ˜ x nI and ˜ x nQ can be in-p hase and quad rature components of two different information v ariables. Since h k is any complex variable, from ( 28), linear proce ssing of the received symbols at the relays alon e cann ot con tribute variables of the form ± h k ˜ x n , ± h ∗ k ˜ x ∗ n . Therefo re, S-PDSSDCs can not be co nstructed by lin ear processing o f the receiv ed symb ols at the relays alone wh en the sour ce transmits the information symbols to all the relay s with o ut precod ing. Remark 3: If h k ’ s are r eal variables, th en Re ( x ) , I m ( x ) ∈ { h k Re ( x n ) , h k Im ( x n ) | 1 ≤ n ≤ N } in which case, the non-zer o entries of the k th row can be of the fo rm ± h k ˜ x n , ± h k ˜ x ∗ n where ˜ x nI and ˜ x nQ can be in-p hase and quadra ture comp onents of two different infor mation variables for any real variable h k . This aspect h as been well studied in [15], [19] and [20] where the r elays are assume d to have the knowledge of phase comp onent of their correspo nding chan nels thereby makin g h k , a r eal variable. Hence, with the k nowledge of partial CSI at th e relays, high rate distributed SSD cod es can b e constructed by linear proce ssing at the r elays alo ne. i.e, with the knowledge of partial CSI at the relays, the source need n ot perform precod ing of information symb ols bef ore tran smitting to the all the relays in the first ph ase. August 10, 2018 DRAFT 18 V I . O N T H E F U L L D I V E R S I T Y O F R S - P D S S D C S In this section, we consider the p roblem of designing a two-dimensional s ignal set, Λ such that a RS-PDSSDC with variables x 1 , x 2 , · · · x N taking values from Λ is fu lly diverse. Since e very codeword of a RS-PDSSDC (Definition 2) contains complex variables h k ’ s, Pairwise error prob ability (PEP) analy sis of RS-PDSSDCs is not straigh tforward. The au thors do no t hav e con ditions on the choice of a com plex sign al set such that a RS-PDSSDC is fully diverse. Howe ver , we m ake the fo llowing con jecture. Conjecture : A RS-PDSSDC in variables x 1 , x 2 , · · · x N is f ully d i verse if th e variables takes values fro m a co mplex signal set say , Λ such that the difference signal s et ∆Λ gi ven by ∆Λ = { a − b | a, b ∈ Λ } does not ha ve any point on the lin es that are ± 45 degrees in the co mplex plane apar t from th e origin. In the r est of this sectio n, we p rovide simu lation results on the perform ance co mparison of a RS-PDSSDC, X (4 , 4) (given in (24)) and a row-monomial DOSTBC, X ′ (4 , 4) (g i ven in (29)) in terms of Symb ol Error Rate (SER) (SER correspon ds to errors in decoding a single comp lex variable). The SER compar ison is provided in Figure 3 . Since the design in (24) has dou ble th e symb ol-rate comp ared to the d esign in (2 9), for a fair comp arison, 16 QAM and a 4 point rotated QPSK are used as signal sets fo r X ′ (4 , 4) and X (4 , 4) respe cti vely to maintain the rate of 1 bits pe r second per Hertz. The average SNR per ch annel use for X ′ (4 , 4) an d X (4 , 4) respectiv ely are 2 p 1 p 2 p 1 +1+2 p 2 and 4 p 1 p 2 p 1 +1+4 p 2 . In ord er to main tain the same Signal to Noise ratio (SNR), for the design X ′ (4 , 4) , every relay (other than the source) uses twice the power as that for the design X (4 , 4) . The class of DOST BCs are shown to b e fully di verse in [1 8]. From Figure 3, it is observed that X (4 , 4) p rovides full d i versity , since th e SER curve moves parallel to that o f X ′ (4 , 4) . It can b e no ticed from Figure 3 that the d esign X (4 , 4) per forms better than X ′ (4 , 4) by close to 2-3 db. X ′ (4 , 4) =         h 1 x 1 h 1 x 2 h 1 x 3 h 1 x 4 0 0 0 0 − h ∗ 2 x ∗ 2 h ∗ 2 x ∗ 1 − h ∗ 2 x ∗ 4 h ∗ 2 x ∗ 3 0 0 0 0 0 0 0 0 h 3 x 1 h 3 x 2 h 3 x 3 h 3 x 4 0 0 0 0 − h ∗ 4 x ∗ 2 h ∗ 4 x ∗ 1 − h ∗ 4 x ∗ 4 h ∗ 4 x ∗ 3         . (29 ) V I I . C O N C L U S I O N A N D D I S C U S S I O N W e consider ed the pro blem o f design ing h igh rate, single-sy mbol decod able DSTBCs when the sou rce is allowed to perform co-ordinate in terleaving of inform ation symbo ls b efore transmitting it to all the relays. W e introduc ed PDSSDCs (Defin ition 2) an d showed th at, DOSTBCs are a special case of PDSSDCs. A specia l class of PDSSDCs h a ving semi-orthog onal pro perty wer e de fined (Defin ition 4). A subset o f S-PDSSDCs called RS-PDSSDCs is studied and an uppe rbound on the m aximal rate of such codes is deriv ed. Th e bounds obtaine d for RS-PDSSDC are shown to be ap proxim ately twice larger than that of DOSTBCs. A systematic constructio n of RS-PDSSDCs are presented fo r the case w hen the numb er of relays, K ≥ 4 . Co des ach ie ving the boun d are found when K is of the form 0 or 3 modulo 4. For the rest of the choices of K , S-PDSSDCs meeting th e above August 10, 2018 DRAFT 19 bound on the rate are no t k nown. T he constructed codes are shown to h av e rate h igher than that of row mono mial DOSTBCs. Some of the possible direc tions for futu re work are a s follows: • In this pap er , w e studied a special class of PDSSDCs called Un itary PDSSDCs (See Defin ition 3). The d esign of high rate Non-Unitary PDSSDCs is an interesting direction f or futu re work . • The authors are n ot aware o f RS-PDSSDCs achieving the b ound on the max imum rate other than the case when K is 0 or 3 modulo 4. 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Hochwald, ”High-Rate codes that are linea r in space and time, ” IEEE T rans. Informati on theory , vol 48, pp. 1804 to 1824, July 2002. LIST OF FIGURES 1. W ireless relay network. 2. V arious class of SSD cod es for coo perative networks. 3. Perfo rmance comp arison o f S-PDSSDC a nd DOSTBC fo r N = 4 and K = 4 with 1 bps/Hz. August 10, 2018 DRAFT 21 . . . Destination Source Relays h h h h g g g g R−1 R−1 1 1 2 2 R R Fig. 1. W irele ss relay network Unitary DSSDC PCIOD PDSSDC S−PDSSDC Non−Unitary DSSDC DSSDC DOSTBC Fig. 2. V arious class of SSD codes for cooperati ve networks 5 10 15 20 25 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR in db SER DOSTBC S−PDSSD Fig. 3. Performance comparison of S-PDS SDC and DOST BC for N = 4 and K = 4 with 1 bps/Hz August 10, 2018 DRAFT